Singularities of implicit differential systems and their integrability

Abstract

Introduction.If P is a smooth manifold and V : P → T P is a vector field on P , i.e. a section of the tangent bundle π 1 : T P → P , then the local integrability of V is a characteristic property, especially for smooth sections V . Traditionally the local integrability of a vector field V (not necessary continuous) on P is defined as an existence at each point p of the domain of V a C

Symplectic Singularities and Solvable Hamiltonian Mappings

“…This notion was generalized in [11,5] Brought to you by | MIT Libraries Authenticated Download Date | 5/9/18 6:31 AM to smooth submanifolds of tangent bundle with possible singular projection into the base space.…”

“…[5]). If π is a tangent bundle projection then the necessary solvability condition pẋ,ẏq P dpπ| M q px,y,ẋ,ẏq pT px,y,ẋ,ẏq M q at px, y,ẋ,ẏq P M is called a tangential solvability condition and extended to the general smooth mapping F " pf, g,ḟ ,ġq : pU, 0q Ñ T R 2n is written in the form pḟ ,ġqpu, vq P JF pu, vqpR 2n q,…”

“…An implicit Hamiltonian system is a solvable isotropic embedding F : R 2n Ą U Ñ T R 2n into the tangent bundle T R 2n endowed with a symplectic structurė ω defined by the canonical flat morphism between tangent and co-tangent bundles of the symplectic space pR 2n , ωq, (see [15]). The solvability properties of F pU q were partially investigated in [5]. In this paper, we extend the notion of implicit Hamiltonian system allowingF to be singular (see [2,12]).…”

“…Thus, we immediately have (cf. [5]) that an implicit differential equation M " F pU q of T R m given by an embedding F " pF ,Ḟ q : U Ñ T R m is smoothly solvable if and only if (1.2) has a smooth solution apuq " pa 1 puq, . .…”

“…(see [5]) Suppose that (1.2) has a solution apuq at every point u P U . If the ideal xdet JF puqy has property of zeros (i.e.…”